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  <meta name="description" content="【数据结构】手写平衡二叉树（AVL）  积千里跬步，汇万里江河。每天进步一点点，终有一天将成大佬 本文源代码：手写AVL树  什么是平衡二叉树？平衡二叉树，又称为AVL树，当树不是空树时，它的左右两个子树的高度差的绝对值不超过1，并且左右两个子树都是一棵平衡二叉树。AVL树查找的时间复杂度为O(logN)。 平衡二叉树基本特点 左右子树深度差不能大于1 左边子树永远比根节点小 右边子树永远比根节点">
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          【数据结构】手写平衡二叉树（AVL）
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        <h1 id="【数据结构】手写平衡二叉树（AVL）"><a href="#【数据结构】手写平衡二叉树（AVL）" class="headerlink" title="【数据结构】手写平衡二叉树（AVL）"></a>【数据结构】手写平衡二叉树（AVL）</h1><p><img src="http://cdn.mjava.top/blog/20200201165139.jpg" alt></p>
<blockquote>
<p>积千里跬步，汇万里江河。每天进步一点点，终有一天将成大佬</p>
<p>本文源代码：<a href="https://github.com/z573419235/studyDemo/blob/master/BaseJava/src/dataStructure/AVLTree.java" rel="noopener external nofollow noreferrer" target="_blank">手写AVL树</a></p>
</blockquote>
<h2 id="什么是平衡二叉树？"><a href="#什么是平衡二叉树？" class="headerlink" title="什么是平衡二叉树？"></a>什么是平衡二叉树？</h2><p>平衡二叉树，又称为AVL树，当树不是空树时，它的左右两个子树的高度差的绝对值不超过1，并且左右两个子树都是一棵<a href="https://baike.baidu.com/item/平衡二叉树/10421057" rel="noopener external nofollow noreferrer" target="_blank">平衡二叉树</a>。AVL树查找的时间复杂度为O(logN)。</p>
<h3 id="平衡二叉树基本特点"><a href="#平衡二叉树基本特点" class="headerlink" title="平衡二叉树基本特点"></a>平衡二叉树基本特点</h3><ul>
<li>左右子树深度差不能大于1</li>
<li>左边子树永远比根节点小</li>
<li>右边子树永远比根节点大</li>
</ul>
<h3 id="平衡二叉树基本结构及操作"><a href="#平衡二叉树基本结构及操作" class="headerlink" title="平衡二叉树基本结构及操作"></a>平衡二叉树基本结构及操作</h3><ul>
<li>左左结构——右旋</li>
</ul>
<p><img src="http://cdn.mjava.top/blog/20200201153234.jpg" alt="左左结构"></p>
<ul>
<li>右右结构——左旋</li>
</ul>
<p><img src="http://cdn.mjava.top/blog/20200201153301.jpg" alt="右右结构"></p>
<ul>
<li>左右结构——左子先左旋，然后整体右旋</li>
</ul>
<p><img src="http://cdn.mjava.top/blog/20200201153329.jpg" alt="左右结构"></p>
<ul>
<li>右左结构——右子先右旋，然后整体左旋</li>
</ul>
<p><img src="http://cdn.mjava.top/blog/20200201153403.jpg" alt="右左结构"></p>
<h2 id="代码实现"><a href="#代码实现" class="headerlink" title="代码实现"></a>代码实现</h2><p>先创建一个内部类Node，来表示树的每个节点</p>
<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">AVLTree</span> </span>&#123;</span><br><span class="line">    <span class="keyword">private</span> Node rootNode;</span><br><span class="line"></span><br><span class="line">    <span class="comment">//二叉树节点</span></span><br><span class="line">    <span class="keyword">private</span> <span class="class"><span class="keyword">class</span> <span class="title">Node</span></span>&#123;</span><br><span class="line">        <span class="keyword">public</span> Node parent; <span class="comment">//父</span></span><br><span class="line">        <span class="keyword">public</span> Node left;	<span class="comment">//左子树</span></span><br><span class="line">        <span class="keyword">public</span> Node right;	<span class="comment">//右子树</span></span><br><span class="line">        <span class="meta">@NotNull</span></span><br><span class="line">        <span class="keyword">public</span> <span class="keyword">int</span> data;	<span class="comment">//存放的数据</span></span><br><span class="line">        <span class="keyword">private</span> <span class="keyword">int</span> depth;	<span class="comment">//深度</span></span><br><span class="line">        <span class="keyword">private</span> <span class="keyword">int</span> balance;	<span class="comment">//平衡因子</span></span><br><span class="line">		<span class="comment">//有参构造方法</span></span><br><span class="line">        <span class="function"><span class="keyword">public</span> <span class="title">Node</span><span class="params">(<span class="keyword">int</span> data)</span></span>&#123;</span><br><span class="line">            <span class="keyword">this</span>.data=data;</span><br><span class="line">            <span class="keyword">this</span>.depth=<span class="number">1</span>;</span><br><span class="line">            <span class="keyword">this</span>.balance=<span class="number">0</span>;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h3 id="插入数据"><a href="#插入数据" class="headerlink" title="插入数据"></a>插入数据</h3><p>暴露一个方法给外部调用</p>
<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="comment">/**添加数据方法*/</span></span><br><span class="line"><span class="function"><span class="keyword">public</span> <span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> data)</span></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (<span class="keyword">this</span>.rootNode==<span class="keyword">null</span>)&#123;</span><br><span class="line">        <span class="keyword">this</span>.rootNode=<span class="keyword">new</span> Node(data);</span><br><span class="line">    &#125;<span class="keyword">else</span> &#123;</span><br><span class="line">        <span class="keyword">this</span>.insert(rootNode,data);</span><br><span class="line">        <span class="comment">//判断根节点是否有父  有的话说明有旋转操作，更新根节点</span></span><br><span class="line">        <span class="keyword">if</span> (<span class="keyword">this</span>.rootNode.parent!=<span class="keyword">null</span>)&#123;</span><br><span class="line">            <span class="keyword">this</span>.rootNode=<span class="keyword">this</span>.rootNode.parent;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<p>实际内部是调用另一个<code>insert</code>方法：</p>
<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="function"><span class="keyword">private</span> <span class="keyword">void</span> <span class="title">insert</span><span class="params">(Node root,<span class="keyword">int</span> data)</span></span>&#123;</span><br><span class="line">    <span class="comment">//插入的数据比根小</span></span><br><span class="line">    <span class="keyword">if</span> (data&lt;root.data)&#123;</span><br><span class="line">        <span class="keyword">if</span> (root.left==<span class="keyword">null</span>)&#123;</span><br><span class="line">            root.left=<span class="keyword">new</span> Node(data);</span><br><span class="line">            root.left.parent=root;</span><br><span class="line">        &#125;<span class="keyword">else</span> &#123;</span><br><span class="line">            <span class="keyword">this</span>.insert(root.left,data);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">//插入的数据比根大</span></span><br><span class="line">    <span class="keyword">if</span> (data&gt;root.data)&#123;</span><br><span class="line">        <span class="keyword">if</span> (root.right==<span class="keyword">null</span>) &#123;</span><br><span class="line">            root.right=<span class="keyword">new</span> Node(data);</span><br><span class="line">            root.right.parent=root;</span><br><span class="line">        &#125;<span class="keyword">else</span>&#123;</span><br><span class="line">            <span class="keyword">this</span>.insert(root.right,data);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    root.balance=<span class="keyword">this</span>.getBalance(root);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root.balance&gt;<span class="number">1</span>)&#123;</span><br><span class="line">        <span class="comment">//判断左子的平衡因子</span></span><br><span class="line">        <span class="keyword">if</span> (root.left.balance&lt;<span class="number">0</span>)&#123;</span><br><span class="line">            <span class="keyword">this</span>.leftTurn(root.left);</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">this</span>.rightTurn(root);</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">if</span> (root.balance&lt;-<span class="number">1</span>)&#123;</span><br><span class="line">        <span class="comment">//判断右子的平衡因子</span></span><br><span class="line">        <span class="keyword">if</span> (root.right.balance&gt;<span class="number">0</span>)&#123;</span><br><span class="line">            <span class="keyword">this</span>.rightTurn(root.right);</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">this</span>.leftTurn(root);</span><br><span class="line">    &#125;</span><br><span class="line">    root.depth=<span class="keyword">this</span>.getDepth(root);</span><br><span class="line">    root.balance=<span class="keyword">this</span>.getBalance(root);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h3 id="右旋"><a href="#右旋" class="headerlink" title="右旋"></a>右旋</h3><blockquote>
<p>右旋的操作如下</p>
<ul>
<li>我父变成左子的父</li>
<li>左子变成我的父</li>
<li>我变成左子的右子</li>
<li>左子的右子变成我的左子</li>
<li>(当左子的右子存在时)我变成左子的右子的父</li>
<li>计算左右节点的深度</li>
<li>计算深度差</li>
</ul>
</blockquote>
<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="function"><span class="keyword">private</span> <span class="keyword">void</span> <span class="title">rightTurn</span><span class="params">(@NotNull Node node)</span></span>&#123;</span><br><span class="line">    Node parent=node.parent;</span><br><span class="line">    Node leftSon=node.left;</span><br><span class="line">    Node leftSon_rightSon=leftSon.right;</span><br><span class="line"></span><br><span class="line">    <span class="comment">//如果父不为空，判断我是在父的左节点还是右节点</span></span><br><span class="line">    <span class="keyword">if</span> (parent!=<span class="keyword">null</span>)&#123;</span><br><span class="line">        <span class="keyword">if</span> (node==parent.left)&#123;</span><br><span class="line">            <span class="comment">//我在父的左节点上，把我的左子变成父的左子</span></span><br><span class="line">            parent.left=leftSon;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span> (node==parent.right)&#123;</span><br><span class="line">            <span class="comment">//我在父的右节点上，把我的左子变成父的右子</span></span><br><span class="line">            parent.right=leftSon;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    leftSon.parent=parent;</span><br><span class="line">    node.parent=leftSon;</span><br><span class="line">    leftSon.right=node;</span><br><span class="line">    node.left=leftSon_rightSon;</span><br><span class="line">    <span class="comment">//如果左子的右子确实存在的</span></span><br><span class="line">    <span class="keyword">if</span> (leftSon_rightSon!=<span class="keyword">null</span>)&#123;</span><br><span class="line">        <span class="comment">//我变成左子的右子的父</span></span><br><span class="line">        leftSon_rightSon.parent=node;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">//重新计算深度和平衡因子</span></span><br><span class="line">    node.depth=<span class="keyword">this</span>.getDepth(node);</span><br><span class="line">    node.balance=<span class="keyword">this</span>.getBalance(node);</span><br><span class="line">    leftSon.depth=<span class="keyword">this</span>.getDepth(leftSon);</span><br><span class="line">    leftSon.balance=<span class="keyword">this</span>.getBalance(leftSon);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h3 id="左旋"><a href="#左旋" class="headerlink" title="左旋"></a>左旋</h3><blockquote>
<p>左旋的操作如下</p>
<ul>
<li>我的父变右子的父</li>
<li>右子变成我的父</li>
<li>我变成右子的左子</li>
<li>右子的左子变成我的右子</li>
<li>(当右子的左子存在时)我变成右子的左子的父</li>
<li>计算左右节点的深度</li>
<li>计算深度差</li>
</ul>
</blockquote>
<figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="function"><span class="keyword">private</span> <span class="keyword">void</span> <span class="title">leftTurn</span><span class="params">(@NotNull Node node)</span></span>&#123;</span><br><span class="line">    Node parent=node.parent;</span><br><span class="line">    Node rightSon=node.right;</span><br><span class="line">    Node rightSon_leftSon=rightSon.left;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (parent!=<span class="keyword">null</span>)&#123;</span><br><span class="line">        <span class="keyword">if</span> (node==parent.left)&#123;</span><br><span class="line">            parent.left=rightSon;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span> (node==parent.right)&#123;</span><br><span class="line">            parent.right=rightSon;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    rightSon.parent=parent;</span><br><span class="line">    node.parent=rightSon;</span><br><span class="line">    rightSon.left=node;</span><br><span class="line">    node.right=rightSon_leftSon;</span><br><span class="line">    <span class="keyword">if</span> (rightSon_leftSon!=<span class="keyword">null</span>)&#123;</span><br><span class="line">        rightSon_leftSon.parent=node;</span><br><span class="line">    &#125;</span><br><span class="line">    node.depth=<span class="keyword">this</span>.getDepth(node);</span><br><span class="line">    node.balance=<span class="keyword">this</span>.getBalance(node);</span><br><span class="line">    rightSon.depth=<span class="keyword">this</span>.getDepth(rightSon);</span><br><span class="line">    rightSon.balance=<span class="keyword">this</span>.getBalance(rightSon);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h3 id="计算深度"><a href="#计算深度" class="headerlink" title="计算深度"></a>计算深度</h3><figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="comment">/**计算深度*/</span></span><br><span class="line"><span class="function"><span class="keyword">private</span> <span class="keyword">int</span> <span class="title">getDepth</span><span class="params">(Node node)</span></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> depth = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">if</span>(node.left==<span class="keyword">null</span> &amp;&amp; node.right!=<span class="keyword">null</span>) &#123;</span><br><span class="line">        depth=node.right.depth;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">if</span>(node.right==<span class="keyword">null</span> &amp;&amp; node.left!=<span class="keyword">null</span>) &#123;</span><br><span class="line">        depth=node.left.depth;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">if</span> (node.right!=<span class="keyword">null</span> &amp;&amp; node.left!=<span class="keyword">null</span>) &#123;</span><br><span class="line">        depth=Math.max(node.left.depth,node.right.depth);</span><br><span class="line">    &#125;</span><br><span class="line">    depth++;</span><br><span class="line">    <span class="keyword">return</span> depth;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h3 id="计算平衡因子"><a href="#计算平衡因子" class="headerlink" title="计算平衡因子"></a>计算平衡因子</h3><figure class="highlight java"><table><tr><td class="code"><pre><span class="line"><span class="comment">/**计算左右深度差*/</span></span><br><span class="line"><span class="function"><span class="keyword">private</span> <span class="keyword">int</span> <span class="title">getBalance</span><span class="params">(Node node)</span></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> leftDepth = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">int</span> rightDepth = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">if</span>(node.left!=<span class="keyword">null</span>)&#123;</span><br><span class="line">        leftDepth=node.left.depth;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">if</span>(node.right!=<span class="keyword">null</span>)&#123;</span><br><span class="line">        rightDepth=node.right.depth;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">/**</span></span><br><span class="line"><span class="comment">         *      左减右</span></span><br><span class="line"><span class="comment">         * 为负数：右边子树高</span></span><br><span class="line"><span class="comment">         * 为正数: 左边子树高</span></span><br><span class="line"><span class="comment">         * */</span></span><br><span class="line">    <span class="keyword">return</span> leftDepth-rightDepth;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<h2 id="附言"><a href="#附言" class="headerlink" title="附言"></a>附言</h2><p>如果代码和静态图看不太明白的话，这边推荐几个动画演示的网站(可能需要科学上网)：</p>
<ul>
<li><p><a href="https://visualgo.net/zh" rel="noopener external nofollow noreferrer" target="_blank">visualgo在线</a></p>
</li>
<li><p><a href="https://www.cs.usfca.edu/~galles/visualization/Algorithms.html" rel="noopener external nofollow noreferrer" target="_blank">数据结构可视化</a></p>
</li>
</ul>

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          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#【数据结构】手写平衡二叉树（AVL）"><span class="nav-number">1.</span> <span class="nav-text">【数据结构】手写平衡二叉树（AVL）</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#什么是平衡二叉树？"><span class="nav-number">1.1.</span> <span class="nav-text">什么是平衡二叉树？</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#平衡二叉树基本特点"><span class="nav-number">1.1.1.</span> <span class="nav-text">平衡二叉树基本特点</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#平衡二叉树基本结构及操作"><span class="nav-number">1.1.2.</span> <span class="nav-text">平衡二叉树基本结构及操作</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#代码实现"><span class="nav-number">1.2.</span> <span class="nav-text">代码实现</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#插入数据"><span class="nav-number">1.2.1.</span> <span class="nav-text">插入数据</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#右旋"><span class="nav-number">1.2.2.</span> <span class="nav-text">右旋</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#左旋"><span class="nav-number">1.2.3.</span> <span class="nav-text">左旋</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#计算深度"><span class="nav-number">1.2.4.</span> <span class="nav-text">计算深度</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#计算平衡因子"><span class="nav-number">1.2.5.</span> <span class="nav-text">计算平衡因子</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#附言"><span class="nav-number">1.3.</span> <span class="nav-text">附言</span></a></li></ol></li></ol></div>
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